Mathematical philosophy, a study of fate and freedom; lectures for educated laymen, by Cassius J. Keyser.

238 MATHEMATICAL PHILOSOPHY
tion, and hence the admissible terms and only these are
important, for it is in virtue of these and only these that
a propositional function is, as we have seen, a matrix,
mould, or source, of propositions. If now we observe that
the admissible terms for a given function <(x) constitute
a type, or class, of terms, we shall be prepared to answer
our question. The answer is: The symbol x in a given
propositional function p(x) is called a variable because the
symbol represents any one of the terms of the class of
admissible terms for O(x) and represents nothing else.
There is nothing subtler in human speech and nothing
more important than the phrase " any one " as here used;
without it, logic, science, philosophy, even the discourse of
the workaday world, would be impossible. What does
the phrase mean? It does not admit of precise definition,
for it is essentially involved (explicitly or implicitly) in
the very act of definition. The only or the best way to
sharpen our sense of its meaning is to meditate upon
examples of its use. A farmer has in his barn three
horses-Black, Sorrel and Gray. He says to his servant:
"John, fetch me a horse from the barn." John asks:
" Which one?" "Any one," replies the farmer. As here
used the phrase "a horse " is a variable because it represents " any one " of a certain class of horses; in representing " any one " of the class, it does not refer to a particular
horse, for evidently " any one" is not a description or
designation of a particular one of the horses; neither does
it refer to all of the horses conjunctively-Black and
Sorrel and Gray-John is not to fetch them all; it does
refer to each of them disjunctively-John is to fetch Black
or Sorrel or Gray-no matter which one. So it is in the
foregoing definition: in representing " any one " of the
class of admissible terms for o(x), x does not refer to a